Positive radial solutions of critical Hénon equations on the unit ball in ℝN$$ {mathbb{R}}^N $$

摘要

In this paper, we study the positive radial solutions for the Hénon equations with weighted critical exponents on the unit ball B$$ B $$ in ℝN$$ {mathbb{R}}^N $$ with N⩾3$$ Ngeqslant 3 $$. We first confirm that 2∗(α)=2(N+α)N−2$$ {2}^{ast}left(alpha right)=frac{2left(N+alpha right)}{N-2} $$ with α>0$$ alpha >0 $$ is the critical exponent for the embedding from H0,r1(B)$$ {H}_{0,r}^1(B) $$ into Lp(B;xα)$$ {L}^pleft(B;{leftxright}^{alpha}right) $$ and name 2∗(α)$$ {2}^{ast}left(alpha right) $$ as the Hénon‐Sobolev critical exponent. Then, following the great ideas of Brezis and Nirenberg (Comm Pure Appl Math. 1983;36:437‐477), we establish the existence and nonexistence of positive radial solutions of the problems with single Hénon‐Sobolev critical exponent and linear or nonlinear but subcritical perturbations. We further study the problems with multiple critical exponents, which may be Hénon‐Sobolev critical exponents, Hardy‐Sobolev critical exponents, or Sobolev critical exponents. The methods and arguments involved with are the mountain pass theorem and the strong maximum principle and the Pohozaev identity.